At a qualitative level, the connection between sum-product estimates, point-line incidences, and Furstenberg sets is well known. In [1]}, Bourgain, Katz, and Tao proved a sum-product estimate over \(\mathbb {F}_p\) , and used this to obtain a point-line incidence bound in \(\mathbb {F}_p^2\) . The argument used by Héra, Shmerkin, and Yavicoli [2]} is also based on these ideas, but uses Bourgain's discretized projection theorem [3]} instead of the discretized sum-product theorem. Our main contribution is that our arguments are carefully structured to obtain an effective quantitative relationship between the discretized sum-product theorem and the size of \((\alpha ,2\alpha )\) -Furstenberg sets. One major difficulty is that while both Theorem REF  and Proposition REF  involve sets that satisfy non-concentration conditions, not all paths between these two non-concentration conditions are equally efficient. To overcome this difficulty, we carefully construct a sequence of point-line configurations that allow us to translate between the two notions of non-concentration. See Figure REF  for an overview of this translation step.
